Warping geometric objects

ABSTRACT

A system is disclosed for warping models made from geometric objects, such as electronic maps, to correct local distortions in the models without compromising model topology. A set of transformation functions are derived from relationships between points in a first model that match points in a second model. The transformation functions are then applied to the points in the first model to generate a new model with reduced distortion. In order to provide for reducing local distortions, warping is applied to selected corresponding regions of the first model and the second model by triangulating these regions and generating transformation functions for each corresponding pair of triangles. Topology preservation is achieved by identifying matching points in the first model and the second model that have a potential for causing topology deviations. Such matching points are then excluded from the process of developing transformation equations to be used in the warping process. Matching points with potential for causing topology deviations are identified by triangulating matching points in the selected regions of the first model and the second model and analyzing the resulting triangles.

CROSS-REFERENCES TO RELATED APPLICATIONS

This Application is related to the following Application:

Matching Geometric Objects, by Hongche Liu and Shyam Kuttikkad, Ser. No.09/176,243, filed Oct. 21, 1998.

The above-cited Application is incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of Invention

The present invention is directed toward the field of warping sets ofgeometric objects.

2. Description of Related Art

Physical areas are often modeled using geometric objects. Examples ofsuch physical areas include a geographic region and a building's floorplan. One example of a model using two-dimensional geometric objects isa map of a geographic region, such as a city.

FIG. 1 illustrates a map that is formed by a set of geometric objects.The objects in FIG. 1 include points 60-70, arcs 80-91, and polygons50-51. Each arc 80-91 represents a road and is bounded at each of itsends by a point. For example, arc 80 is bounded at one end by point 60and at another end by point 62. Each arc can be co-bounded by one or twopolygons. For example, arc 86 is co-bounded by polygons 50 and 51, whilearc 85 is only co-bounded by polygon 50. Each polygon 50 and 51 isformed by a set of arcs that define the boundary of the polygon. Forexample, polygon 50 is defined by boundary arcs 81, 82, 83, 84, 85, and86.

Different models of the same physical area often provide differentquality details about the area. Differences between models of the samearea can occur, because models can be distorted, so that all thefeatures in the model are not depicted in their true positions. Suchdistortions are often localized, with different regions of the modelbeing distorted differently. In paper models, distortion can be causedby paper stretching, folding, and scaling. In electronic models,distortion can be caused when the model is transformed into digitalform.

By repositioning different models of the same area, a more accurate newmodel can be created. FIG. 2 illustrates a map 100 of an area that isneither very smooth nor accurate. FIG. 3 shows a more accurate map 10 ofthe same area as map 100. The differences between maps 100 and 10 areillustrated in FIG. 4, which shows an overlay of maps 100 and 110. Arepositioning process can be performed to utilize the details of bothmaps 100 and 110 to obtain an improved new map.

The use of repositioning is particularly beneficial and practical isgenerating electronic maps. There are often many different mapsavailable for a particular region. By making use of a computer toreposition the many maps that are available for a region, a veryaccurate new map can be generated in a time efficient and cost efficientmanner.

One known method for repositioning two models is to warp one model basedon the weighted sum of local displacements between points in the twomodels. In this method, a set of matching points in the models isobtained for selected regions of the models. Displacements betweenmatching points in each selected region are then employed to generate awarping transformation for the selected region. The warpingtransformation is then used to transform points in one of the modelsinto points in a new model.

However, such a method is inadequate, because it does not provide formaintaining the topology of the original models in the new model.Topology refers to the relative connections and orientations ofgeometric objects in the model, and its preservation is critical toensuring the accuracy of the new model. For example, if topology is notpreserved, arc intersections which do not exist in either of theoriginal models may be inserted into the new model.

In order to better maintain topology, model-based warping can beemployed. One known method of such warping includes the creation of aglobal transformation model that provides for transforming all of thepoints in a model into points in a new model. In such a method, allmatching points from two existing models are employed to determineparameters for the global transformation model using numerical analysis,such as least mean squares techniques. However, this global approachdoes not provide for eliminating local distortions, which constitutemany of the distortions that are encountered.

Accordingly, there is a need for a warping process that can be appliedto models to correct local distortions without compromising topology.

SUMMARY OF THE INVENTION

The present invention, roughly described, provides for warping modelsmade from geometric objects, such as electronic maps, to correct localdistortions in the models without compromising model topology. Thewarping is performed by using a set of transformation functions that arederived from relationships between points in a first model that matchpoints in a second model with more accurate positioning. Thetransformation functions are applied to the points in the first model togenerate a new model with reduced distortion.

In order to provide for reducing local distortions, warping is appliedto selected corresponding regions of the first model and the secondmodel by triangulating these regions and generating transformationfunctions for each corresponding pair of triangles. Topologypreservation is achieved by identifying matching points in the firstmodel and the second model that have a potential for causing topologydeviations. Such matching points are then excluded from the process ofdeveloping transformation equations to be used in the warping process.

Matching points with potential for causing topology deviations areidentified by triangulating matching points in the selected regions ofthe first model and the second model and analyzing the resultingtriangles. A matching point is identified as having potential forcausing topology deviations, if it is a vertex in a triangle in onemodel that has an inverted corresponding triangle in the other model. Amatching point is also identified as having potential for causingtopology deviations, if it is a vertex in a triangle having too small anarea or height. An iterative process is performed to identify suchundesirable matching points.

These and other objects and advantages of the present invention willappear more clearly from the following description in which thepreferred embodiment of the present invention has been set forth inconjunction with the drawings.

BRIEF DESCRIPTIONS OF THE DRAWINGS

The present invention will be described with reference to the followingdrawings, in which:

FIG. 1 illustrates a model using geometric objects.

FIG. 2 is a first map of an area.

FIG. 3 is a second map of the area shown in FIG. 2.

FIG. 4 depicts the map in FIG. 2 being overlaid by the map in FIG. 3.

FIG. 5 illustrates a sequence of operations for repositioning one modelto another.

FIG. 6 illustrates a sequence of operations for identifying matchingarcs in accordance with the present invention.

FIG. 7 illustrates a sequence of operations for identifying matchingpolygons.

FIG. 8 illustrates a sequence of operations for determining the degreeof similarity between a pair of polygons.

FIG. 9 illustrates two sets of polygons for which proximity is beingmeasured.

FIG. 10 illustrates polygons for which area values are being determined.

FIG. 11 illustrates a sequence of operations for assigning attributes tomatching arcs.

FIG. 12 illustrates a sequence of operations for warping a model inaccordance with the present invention.

FIG. 13 illustrates a sequence of operations for identifying matchingpoints that have potential for causing topology deviations.

FIGS. 14 illustrates the results of performing a triangulation on afirst set of matching points and applying the same triangulation to acorresponding second set of matching points.

FIG. 15 illustrates a sequence of operations for initially identifyingmatching points that may cause topology deviations.

FIG. 16 illustrates a sequence of operations for replacing matchingpoints.

FIGS. 17(a)-(b) illustrate the successful replacement of a matchingpoint in one embodiment of the present invention.

FIGS. 18(a)-(b) illustrate the successful replacement of a matchingpoint in an alternate embodiment of the present invention.

FIG. 19 illustrates a sequence of operations for transforming selectedregions of a model.

FIGS. 20(a)-(c) illustrate intersection points.

FIG. 21 illustrates a block diagram of one exemplar hardwarearchitecture that can be used to practice the present invention.

DETAILED DESCRIPTION

FIG. 5 illustrates a sequence of operations for performing arepositioning of two sets of geometric objects, in accordance with thepresent invention. In one embodiment of the present invention, each setof geometric objects is an electronic map. In such an embodiment, therepositioning process is carried out by a computer system which is ableto access both of the electronic maps and a set of instructions thatprovides for repositioning the two electronic maps. Such a computersystem will be described in greater detail below.

In describing aspects of the present invention, maps, and in particularelectronic maps, will be used as an example of a set of geometricobjects. However, one with ordinary skill in the art will recognize thata set of geometric objects can be used to represent a number of otherthings, such as the floor plan of a building, a layout of an integratedcircuit, and a machine drawing. Accordingly, the invention is in no waylimited to electronic maps and can be practiced using any set ofgeometric objects.

As shown in FIG. 5, repositioning is initiated in step 130 byidentifying arcs in a first electronic map (first set of geometricobjects) for which there is a matching arc in a second electronic map(second set of geometric objects). The matching arcs are then employedin step 132 to identify points in the first map that match points in thesecond map in step 132. Next, geometric objects in the first map arewarped to create a new map, based on a transformation function that isderived from a set of the matching points identified in step 132.

In embodiments of the present invention, the arc matching performed instep 130 is based on geometry and topology information associated withthe maps. This avoids any reliance on arc attributes, which may not besupplied. In general, arc matching is performed by identifying arcs inthe first map that are co-bounded on both sides by polygons which matchthe corresponding co-bounding polygons of the arc in the second map.

In order to identify matching polygons, a set of similarity metrics aredetermined for each pairing of a polygon in the first map and a polygonin the second map. The similarity metrics are then employed to determinethe degree of similarity between polygons and whether or not they match.In one embodiment of the present invention, the similarity metricsinclude isolated metrics that indicate the similarity of the shape,proximity, area, and rotation of polygons being compared.

The identification of matching points in step 132 is based on thematching arcs identified in step 130 and can be performed in a number ofways that are well known in the art. One such process for identifyingmatching points is disclosed in U.S. Pat. No. 5,546,107, entitledAutomatic Chain-Based Conflation of Digital Maps. However, many otherprocesses can be employed.

In accordance with the present invention, the warping of the first mapin step 134 is performed so that local distortions in the first andsecond maps are removed without compromising the topology represented inthe two maps. In one embodiment of the present invention, selectedcorresponding regions of each map are employed in the warping process.For the selected regions, it is determined which matching points havepotential for causing topology deviations to occur in the new map beingcreated. These points are regarded as outliers and are rejected from theset of matching points. Transformation equations are then derived usingthe remaining matching points. Finally, the transformation equations areused to transform points in the first map to generate the new map.

In order to perform the repositioning process described in FIG. 5,several requirements must be met. The positions and shapes of allgeometric objects (points, arcs, and polygons) must be provided withrespect to a common coordinate system. The origin for each coordinatesystem is to represent the same physical location. All topologicalinformation relating to the maps must also be provided. Such informationincludes the points bounding each arc, the arcs defining each polygon,the arcs connected to each point, and the polygons co-bounding each arc.

When warping, it is also critical that the selected region of interestin each map represents the same physical location. This can be ensuredby having the vertices for a selected region in the first map representthe same real world locations as the vertices for the selected region inthe second map. The need for this requirement will be explained below.

FIG. 6 illustrates a sequence of operations that are performed toidentify matching arcs (step 130, FIG. 5) in the two selected maps.First, matching polygons in the two maps are identified in step 140. Thematching polygons are then employed to designate matching arcs in step142.

FIG. 7 illustrates a sequence of operations employed in one embodimentof the present invention for identifying matching polygons in differentmaps (step 140, FIG. 6). First, a pair of polygons is selected in step150, with one polygon being in the first map and the other polygon beingin the second map. Next, the degree of similarity of the selectedpolygons is determined in step 152. Once the degree of similarity isdetermined, a determination is made of whether there are any unselectedpairs of polygons in step 154. If it is determined that there is a pairof polygons that has not yet been selected, a new pair is selected instep 150 and the above described process is repeated. Otherwise,matching polygons are paired in step 156, based on the similaritydeterminations made in step 152.

FIG. 8 illustrates a sequence of operations that is employed in oneembodiment of the present invention for determining the degree ofsimilarity between two polygons (step 152, FIG. 7). First, a set ofsimilarity metrics for the selected pair of polygons is determined(steps 160, 162, 164, and 166). Next, the degree of similarity betweenthe selected pair of polygons is calculated in step 168, based on theset of similarity metrics. As shown in FIG. 8, similarity metrics areobtained for the relative proximity, area, shape, and rotation of theselected polygons. However, in alternate embodiments of the presentinvention, other similarity metrics, or a subset of the similaritymetrics shown in FIG. 8, or a combination thereof are included in theset of similarity metrics.

As shown in FIG. 8, a proximity similarity metric is first determined instep 160, independent of the area, shape, and rotation metrics. As afirst step in determining the proximity, a centroid is computed for eachpolygon in the selected pair. In order to determine the centroid for ageometric object, N number of points are selected along the polygon'sperimeter. In one embodiment of the present invention, the N number ofpoints are selected with each of the selected points being separated byan equal distance along the polygon's perimeter. The coordinates of thecentroid are then calculated according to the following equation:$\begin{matrix}{C = \left( {\frac{\sum\limits_{i = 1}^{N}x_{i}}{N},\frac{\sum\limits_{i = 1}^{N}y_{i}}{N}} \right)} & \text{Equation 1}\end{matrix}$

wherein:

C is the x and y coordinates for the centroid;

x_(i) is an x coordinate for the ith selected point on the polygon'sperimeter; and

y_(i) is the y coordinate for the ith selected point on the polygon'sperimeter.

Although the invention is described in terms of the Cartesian coordinatesystem (x, y), one with ordinary skill in the art will recognize thatthe use of other coordinate systems is within the scope of the presentinvention.

Once the centroid for each of the selected polygons has been determined,the displacement between the centroids is calculated according to thefollowing equation:

D={square root over ((x _(A) −x _(B))²+(y _(A) −y _(B))²)}  Equation 2

wherein:

D is the displacement;

x_(A) is the x coordinate for the centroid in the polygon from the firstmap;

x_(B) is the x coordinate for the centroid in the polygon from thesecond map;

y_(A) is the y coordinate for the centroid in the polygon from the firstmap; and

y_(B) is the y coordinate for the centroid in the polygon from thesecond map.

Once the displacement has been determined, the proximity similaritymetric is calculated according to the following equation:$\begin{matrix}{{{P = {{\frac{\left( {B - D} \right)}{\left( {B + D} \right)}\quad {if}\quad B} > D}},{else}}{P = 0}} & \text{Equation 3}\end{matrix}$

wherein:

P is the proximity similarity metric; and

B is a normalizing constant.

In one embodiment of the present invention, N is equal to 32, andnormalizing constant B is user selectable based on the quality of themaps being repositioned. In a typical example, the first map has a 1:24Kscale, yielding an accuracy tolerance of 40 feet, and the second map hasa 1:100K scale, yielding an accuracy tolerance of 160 feet. In such anexample, normalizing constant B is equal to the sum of the tolerances ofthe two maps, which is equal to 200 feet.

FIG. 9 illustrates the effective usefulness of the proximity similaritymetric. FIG. 9 includes a first set of polygons 170 and a second set ofpolygons 172. Each of the polygons in sets 170 and 172 are similar inarea, rotation, and shape. If the proximity similarity metric isdetermined individually for each combination of polygon 174 in set 170and each of the polygons in set 172, then the proximity similaritymetric will be best for polygon 176. This is because the centroid ofpolygon 176 has the smallest displacement from the centroid of polygon174.

Once the proximity similarity metric has been determined (step 160, FIG.8), an area similarity metric is determined in step 162. First, N numberof preliminary vectors are generated for each of the selected polygons.Each polygon's preliminary vectors are generated to extend from thepolygon's centroid to each of the N points that were selected along thepolygon's perimeter in determining the proximity similarity metric.

Next, a vector set is established for each polygon. For each polygon,the polygon's centroid is subtracted from the start point and end pointof each preliminary vector in the polygon. As a result, the vectors eachappear to extend from the origin. This removes the influence of thepolygon's location on subsequent metrics.

FIG. 10 illustrates sets of vector generated for a pair of polygons 180,181. Vectors V_(A1)-V_(AN) have been generated for polygon 180. Each ofvectors V_(A1)-V_(AN) extends from origin 184 to one of N points 182_(1−N) along the perimeter 186 of polygon 180. Vectors V_(B1)-V_(BN)have been generated for polygon 181. Each of vectors V_(B1)-V_(BN)extends from origin 184 to one of N points 183 _(1−N) along theperimeter 187 of polygon 181.

Next an area value is calculated for each selected polygon, according tothe following equation: $\begin{matrix}{{AV} = {\sum\limits_{i = 1}^{N}{V_{i}}^{2}}} & \text{Equation 4}\end{matrix}$

wherein:

AV is the area value; and

∥V_(i)∥ is the length of an ith one of the vectors in the polygon.

Once an area value has been determined for each polygon, an areasimilarity metric is determined according to the following equation:$\begin{matrix}{A = \frac{{MIN}\left( {{AV}_{A},{AV}_{B}} \right)}{{MAX}\quad \left( {{AV}_{A},{AV}_{B}} \right)}} & \text{Equation 5}\end{matrix}$

wherein:

A is the area similarity metric;

AV_(A) is the area value for the polygon in the first map;

AV_(B) is the area value for the polygon in the second map;

MIN( ) is an instruction indicating that the lowest value appearingwithin the parentheses is to be selected; and

MAX( ) is an operation indicating that the maximum value appearingbetween the parentheses is to be selected.

Accordingly, the area similarity metric will represent the ratio of thesmallest area value to the largest area value for the two selectedpolygons.

Once the area similarity metric has been determined (step 162, FIG. 8),a shape similarity metric is determined in step 164. In order todetermine the shape similarity metric, each of the vectors generated ineach selected polygon is normalized according to the following equation:$\begin{matrix}{V_{norm} = \left( {\frac{x}{\sqrt{AV}},\frac{y}{\sqrt{AV}}} \right)} & \text{Equation 6}\end{matrix}$

wherein:

V_(norm) is the normalized vector;

x is the vector's x-axis component;

y is the vector's y-axis component; and

AV is the area value for the polygon in which the vector resides.

The normalization discounts any area difference between the polygons,thereby allowing the shape determination to be made in isolation.

Next, a set of shape measures are obtained as the sum of the squares ofvector differences between the two selected polygons' sets of normalizedvectors at different rotations. The minimum shape measure value that isthen designated as the shape similarity metric.

A shape measure for the ith rotation of the polygons is determinedaccording to the following equation: $\begin{matrix}{{SM}_{i} = {\sum\limits_{j = 1}^{N}{{\overset{\_}{V_{Aj}} - \overset{\_}{V_{B{({j + i})}}}}}^{2}}} & \text{Equation 7}\end{matrix}$

wherein:

SM_(i) is the ith shape measure;

{overscore (V_(Aj))} is the jth normalized vector in the polygon fromthe first map; and

{overscore (V_(B(j+i)))} is the (j+i)th normalized vector in the polygonfrom the second map, wherein (j+i) is equal to the sum of j and i if thesum is less than or equal to N, else (j+i) is equal to the sum of j andi minus N.

The shape similarity metric is the minimum SM_(i) value that isdetermined for i spanning a range from 0 to (N−1). By allowing i to spanfrom 0 to (N−1), all of the possible shape measures between the twopolygons are made.

In order to determine the shape similarity metric, the followingequation is employed:

S=MIN(SM _(i) for 0≦i≦(N−1))  Equation 8

wherein:

S is the shape similarity metric.

Once the shape similarity metric has been determined (step 164, FIG. 8),a rotation similarity metric is determined in step 166. Using therotation i value that yielded the minimum shape measure value (SM_(i)) arotation measure is made of the angles between the corresponding vectorsin the two polygons. The corresponding vectors are those that weresubtracted (Equation 7) from each other when determining the shapemeasure value (SM_(i)) that was selected to be used as the shapesimilarity metric.

In one embodiment of the present invention, the rotation measure iscalculated according to the following equation: $\begin{matrix}{\theta = {\frac{\sum\limits_{j = 1}^{N}{\angle \quad \left( {V_{Aj} - V_{B{({j + i})}}} \right)}}{N}}} & \text{Equation 9}\end{matrix}$

wherein:

θ is the rotation measure;

∠(V_(Aj)−V_(B(j+i))) is an operation indicating that a measure is takenof the angle between the V_(Aj) vector and the V_(B(j+i)) vector;

V_(Aj) is the jth vector in the selected polygon from the first map;

V_(B(j+i)) is the (j+i)th vector in the selected polygon from the secondmap, wherein i is equal to the i value that yielded the shape measurevalue (SM_(i)) that was selected as the shape similarity metric, andwherein (j+i) is the same as described above with reference to Equation7.

In one embodiment of the present invention, the angle between thevectors is determined by taking the arc tangent of the two vectors(V_(Aj) and V_(B(j+i))).

Once the rotation measure is obtained, it is used to calculate arotation similarity metric according to the following equation:$\begin{matrix}{{{R = {{\frac{\left( {\Phi - \theta} \right)}{\left( {\Phi + \theta} \right)}\quad {if}\quad \Phi} > \theta}},{else}}{R = 0}} & \text{Equation 10}\end{matrix}$

wherein:

R is the rotation similarity metric; and

Φ is a normalizing constant.

In one embodiment of the present invention, Φ is equal to 45°.

Once the rotation similarity metric is determined (step 166, FIG. 8), amatching metric is calculated for the selected pair of geometric objectsin step 168. The matching metric is calculated by multiplying theproximity similarity metric (P), area similarity metric (A), shapesimilarity metric (S), and rotation similarity metric (R). This willyield a value ranging from negative infinity to 1, with 1 being thestrongest level of matching.

As described above with reference to FIG. 7, the matching metricindicates a degree of similarity between polygons that is employed topair matching polygons in the two maps (step 156, FIG. 7). In order fortwo polygons to be paired as matching, their matching metric must beequal to or greater than a minimum matching threshold. In one embodimentof the present invention, the minimum matching threshold is 0.35.

In some instances, polygons in one map can have matching metrics thatexceed the minimum matching threshold for multiple polygons in the othermap. However, each polygon in a map can only be paired with one otherpolygon in the other map to form a pair of matching polygons. No polygoncan be paired as matching more than one other polygon.

In one embodiment of the present invention, when such multiple highmatching metrics occur, the following rules are applied in pairingpolygons as matching (step 156, FIG. 7):

1. For each polygon in the second map, pair it with the polygon in thefirst map that has the highest matching metric that exceeds the minimummatching threshold. In the case of a tie, select either one of thepolygons in the first map that tied.

2. If multiple polygons in the second map are then paired to the samepolygon in the first map, then select the pairing with the highestmatching metric. In the case of a tie, select either one of thepairings.

As shown in FIG. 6, the matching polygons are now employed to designatematching arcs in the maps (step 142, FIG. 6). FIG. 11 illustrates asequence of operations for designating the matching arcs. First, an arcin the first map is selected in step 190. Next, a determination is madeof whether the arc is co-bounded on both sides by polygons that havebeen paired with a matching polygon in the second map (step 191).

If it is determined that the selected arc is co-bounded on both sides bypolygons that have been paired with matching polygons in the second map,then the arc is assigned an attribute in step 192. The assignedattribute can then be employed in a process for identifying nodes in thefirst map that match nodes in the second map. In one embodiment of thepresent invention, the assigned attribute is a label formed by aconcatenation of assigned unique identifiers for each co-boundingmatched polygon. In an alternate embodiment, the assigned attribute is alabel, such as a street name.

If the selected arc is determined not to be bounded on both sides bypolygons that have been paired with matching polygons in the second map,or once an arc is assigned an attribute in step 192, then it isdetermined in step 194 whether any of the arcs in the first map have notyet been selected. If there are unselected arcs, then a new arc isselected in step 190 and the above-described process is repeated.Otherwise, the process of designating matching arcs is completed.

Once matching arcs have been identified (step 130 FIG. 5), matchingpoints in the two maps are identified. As described above with referenceto FIG. 5 (step 132), processes for identifying matching points are wellknown in the art.

In particular, a process described in U.S. Pat. No. 5,546,107 can beemployed to identify such matching points by utilizing matching arcs forthe two maps.

Once matching points in the two maps are identified (step 132, FIG. 5),the warping process (step 134) as described above with reference to FIG.5 is initiated. FIG. 12 illustrates a sequence of operations that areperformed in order to effectuate the warping in one embodiment of thepresent invention. First, a corresponding pair of regions are identifiedin the first map and the second map in step 200. As described above,these regions must correspond to the same physical area in each map. Theimportance of this requirement will be explained in greater detailbelow. By performing the warping on selected regions, local distortionsin the first and second maps are effectively corrected, without beingaffected by distortions in other regions.

Once the regions are selected, the matching points in the selectedregions that have potential for causing a topology deviation in the newmap that is to result from warping are identified in step 201. Anexample of such a topology deviation is an inversion of geometricobjects. These points are identified, so that they will not beincorporated in the creation of transformation equations that are usedin the warping process. A process for identifying such points in oneembodiment of the present invention will be described in greater detailbelow.

After the point identification process in step 201 is complete, theselected region in the first map is transformed to form a new map instep 216. This transformation is performed using a set of equation thatare derived from the relationship between corresponding matching pointsin the first and second maps, except for the points identified in step201. A process for performing such a transformation in one embodiment ofthe present invention will be described in greater detail below.

After the transformation of the selected region in the first map (step216) is done, a determination is made in step 217 of whether morewarping is desired. If more warping is desired, then a new pair ofregions is selected in step 200 and the above-described process stepsare updated. Otherwise, the warping is done.

FIG. 13 illustrate a sequence of operation performed in one embodimentof the present invention for identifying matching point having potentialfor causing topology deviations (step 201, FIG. 12). First, the selectedregion in the first map is triangulated in step 202. A triangulation isperformed by forming a set of triangles within the region using thematching points in the region and the boundary points that define thevertices of the region selected for warping. For purposes of thisApplication, these boundary points will also be referred to as matchingpoints. In triangulation, one edge is drawn between each matching pointin the selected region of the first map, such that each edge belongs toeither two triangles or one triangle and the area outside of theselected region.

In one embodiment of the present invention, a Delaunay triangulation isperformed. For each set of points in a two-dimensional plane, thereexists a unique Delaunay triangulation. In a Delaunay triangulation, thetriangles that are formed are all made to be as close as possible tobeing equilateral triangles. In a Delaunay triangulation of a set ofpoints, no point in the set of points falls in the interior of a circlethat passes through all three vertices of any triangle in thetriangulation. Methods for performing Delaunay triangulation are wellknown in the art and in fact are prior art with respect to thisApplication.

A further explanation of Delaunay triangulation can be found in thefollowing references:

“Mesh Generation and Optimal Triangulation” by Marshall Bern and DavidEpstein, which was published in Computing and Euclidean Geometry,Ding-Zhu Du and Frank Hwang editors, World Scientific, Singapore, pp.23-90, 1992;

Jonathan Richard Shewchuk, “Triangle: Engineering a 2D Quality MeshGenerator and Delaunay Triangulator”, First Workshop on AppliedComputational Geometry (Philadelphia, Pennsylvania) pages 124-133, ACM,May 1996; and

Jim Ruppert, “A Delaunay Refinement Algorithm for Quality 2-DimensionalMesh Generation”, Journal of Algorithms 18(3): 548-585, May 1995.

The above-cited references are hereby incorporated by reference. Furtherinformation concerning Delaunay triangulation can be found at thefollowing location on the Internet:http://www.cs.cmu.edu/˜quake/triangle.html.

Once the triangulation of the selected region in the first map iscompleted, the same triangulation is applied to the selected region inthe second map in step 202. In applying the triangulation to the secondmap, a new triangulation process is not executed. Instead, edges areformed between the same matching points in the selected region in thesecond map as were formed between the corresponding matching points inthe selected region in the first map.

FIG. 14 illustrates the results of the triangulation of a selectedregion in the first map and a selected region in the second map, inaccordance with the present invention. In FIG. 14, a set 240 of pointsP₁-P₆ are all of the matching points in the selected region of the firstmap. A triangulation of these points could result in the followingtriangles being formed (Please note that these triangles are not drawnto scale for the purpose of demonstrating a Delaunay triangulation):

(P₁, P₆, P₅)

(P₅, P₆, P₄)

(P₆, P₄, P₃)

(P₆, P₃, P₂)

(P₁, P₂, P₆).

A set 242 of points P₁′-P₆′ are the matching points in the selectedregion of the second map that match points P₁-P₆, respectively. As shownin FIG. 14, the same triangles that were formed using points P₁-P₆ areformed in the selected region of the second map to form the followingcorresponding triangles:

(P₁′, P₆′, P₅′)

(P₅′, P₆′, P₄′)

(P₆′, P₄′, P₃′)

(P₆′, P₃′, P₂′)

(P₁′, P₂′, P₆′).

Once the triangulation is completed for both of the selected regions, aninitial identification of topology deviating matching points isperformed in step 206. This determination is made in one embodiment bycomparing each triangle in the selected region of the first map to acorresponding triangle in the selected region of the second map. In onesuch embodiment, the comparison is made to determine whether thetriangles being compared are inverted. Such an inversion indicates thatthere is a potential for topology deviations. The fact that an inversionidentifies potential for topology deviation is a result of the selectedregions corresponding to the same physical area in the first and secondmaps, as described above.

FIG. 15 illustrates a sequence of operations performed in a accordancewith the present invention to make an initial identification of whethermatching points in the selected regions of the first and second mapshave potential for causing topology deviations (step 206, FIG. 13).First, a pair of triangles is selected in step 230. The selected pairincludes a triangle from the selected region of the first map and atriangle from the selected region of the second map, wherein thetriangle in the second map has vertices that are matching points of thevertices of the triangle in the first map.

Next, the selected triangles are analyzed to determine whether thematching points making up the vertices of the triangles have potentialfor causing topology deviations. One aspect of this determination isdetermining whether the triangles are inverted from one another. Asdescribed above, such an inversion signals the existence of a matchingpoint with potential for causing topology deviations.

FIG. 14 illustrates a pair of inverted triangles. As described above,FIG. 14 shows a set 240 of triangulated matching points (P₁-P₆) for aselected region in the first map and a set 242 of triangulated matchingpoints (P₁′-P₆′) for a corresponding selected region in the second map.Triangle (P₆, P₃, P₂) and corresponding triangle (P₆′, P₂′, P₃′) areinverted with respect to one another. These triangles are considered tobe inverted, because the P₆ and P₆′ points lie on opposite sides of theedge passing between P₃, P₂ and P₂′, P₃′.

In order to determine whether triangles are inverted in one embodimentof the present invention, cross product calculations are employed. Across product is taken of any two edges in the selected triangle in thefirst map, and a cross product is taken of the same two edges in thecorresponding triangle in the second map. For example, with reference toFIG. 14, the cross products could be taken of the P₃, P₆ edge and P₃, P₂edge in set 240 and the P₃′, P₆′ edge and P₃′, P₂′ edge in set 242.

In taking the cross product of two edges, each edge is treated as avector extending from the point where the edges intersect. For example,the P₃, P₆ edge and P₃, P₂ edge are treated as a first vector from P₃ toP₆ and a second vector from P₃ to P₂. When taking the cross product ofthese vectors, the following equation is employed:

CP=(x _(e1) −x _(s1))×(y _(e2) −y _(s2))−(x _(e2) -x_(e2))×(y_(e1)-y_(s1))  Equation 11

wherein:

CP is the cross product;

x_(e1) is the x coordinate of the end point of the first vector;

x_(s1) is the x coordinate of the start point of the first vector;

y_(e1) is the y coordinate of the end point of the first vector;

y_(s1) is the y coordinate of the start point of the first vector;

x_(e2) is the x coordinate of the end point of the second vector;

x_(s2) is the x coordinate of the start point of the second vector;

y_(e2) is the y coordinate of the end point of the second vector; and

y_(s2) is the y coordinate of the start point of the second vector.

If cross products for both of the triangles being compared have the samesign (i.e. positive-positive or negative-negative), then there is noinversion. However, if the cross products are of opposite signs, such aspositive-negative or negative-positive, then the triangles are invertedand both considered to contain matching points that have potential forcausing topology deviations. If either of the cross products is 0, thenthe matching points associated with both triangles are also consideredto have a potential for causing topology deviations.

In addition to determining whether triangles are inverted in step 232(FIG. 15), in one embodiment of the present invention, triangledimensions are also evaluated. For example, if either triangle in aselected pair of corresponding triangles does not have at least aminimum area, then the associated matching points for both triangles areconsidered to have the potential for causing topology deviations.Additionally, if either triangle of a corresponding pair of trianglesdoes not have at least a minimum height, as its smallest height, thenthe matching points for both triangles are considered to have potentialfor causing topology deviations. In one such embodiment, the minimumarea threshold is 15 square inches, and the minimum height threshold is3 inches. Testing for minimum height and area accounts for potentialproblems with precision on a digital computer. Without this test, atriangle with all its contents can be potentially warped to become atiny triangle, which may not have enough room to hold all the contentsof the original triangle.

Once the selected triangles have been analyzed (step 232, FIG. 15), adetermination is made of whether any pair of triangles has not yet beenselected in step 236 (FIG. 15). If it is determined that there is a pairof triangles that has not yet been selected, then a new pair oftriangles is selected in step 230 and the above-described process isrepeated. Otherwise, the initial identification of matching points thathave potential for causing topology deviations is completed.

Once the matching points with potential for causing topology deviationshave been identified (step 206, FIG. 13), a determination is made instep 208 (FIG. 13) of whether any such points were in fact identified inthe last time that step 206 was executed. If it is determined that suchundesirable matching points were identified, then a set of theseundesirable matching points lose their status as matching points and areplaced in a bad point queue in step 210. When a matching point loses itsmatching status, any triangle having the point as a vertex is consideredto be removed. In one embodiment of the present invention, the matchingpoints that lose their status are the matching points that were mostfrequently identified as having potential for causing topologydeviations the last time that step 206 was executed. In an alternateembodiment, points continue to lose matching status, until all invertedor undesirably dimensioned triangles are removed.

After matching status is removed, the selected region from the first mapand the selected region in the second map are triangulated again insteps 202 and 204, respectively, using only the matching points thathave not lost their matching point status. The above-described initialidentification step (206) is then repeated.

If it is determined in step 208 that no matching points were identifiedas having potential for causing topology deviations in the last timethat step 206 was executed, then a determination is made of whether thebad point queue is empty in step 212. The bad point queue is consideredto be empty when it contains either no points or only points that areincapable of regaining matching status, as will be described below. Ifthe bad point queue is not empty, then an attempt is made to replace apoint in the queue in the selected regions in step 214. If the point canbe successfully replaced into the selected regions, so that it no longeris considered to have potential for causing a topology deviation, thenthe point's status as a matching point is returned. A method forattempting such a replacement will be described below.

Once a replacement attempt (step 214) has been made, it is once againdetermined whether or not the bad point queue is empty in step 212. Ifit is determined that the bad point queue is empty, then the process ofidentifying matching points with potential for causing topologydeviations (step 201, FIG. 12) is done.

FIG. 16 illustrates a sequence of operations for attempting to replace apoint from the bad point queue into the selected regions (step 214, FIG.13). First, a point in the bad point queue is selected in step 250.Next, a determination is made of whether the point resides withincorresponding triangles in the selected region in the first map and theselected region in the second map (step 252). FIG. 17(a) illustrates asituation in which point 272 resides within both triangle 270 withvertices (P₁, P₂, P₃) in the selected region of the first map andcorresponding triangle 274 with vertices (P₁′, P₂′, P₃′) in the selectedregion of the second map.

If it is determined that the selected point does not reside withincorresponding triangles, then it is determined in step 254 whether theselected point resides within adjoining triangles in the selectedregions of the first and second maps. Such a circumstance of a pointresiding in adjoining triangles is shown in FIG. 18(a). In the selectedregion in the first map, point 294 resides within triangle 292 withvertices (P₁, P₂, P₃), which adjoins triangle 290 with vertices (P₄, P₂,P₃). In the selected region in the second map, point 294 resides withintriangle 296, which has vertices (P₄′, P₂′, P₃′) and corresponds totriangle 290. Triangle 296 is also adjoined by triangle 298 whichcorresponds to triangle 292 and has vertices (P₁′, P₂′, P₃′). If it isdetermined in step 254 that the selected point does not reside inadjoining regions, then the point is not replaced, and the replacementattempt is done.

If it is determined in step 252 that the point resides in correspondingtriangles in the selected regions of the first and second maps, then anew triangulation is performed in each of the corresponding trianglesusing the selected point in step 260. In each corresponding triangle,the point is connected by an arc to each vertex of the triangle toachieve the new triangulation. An example of such a triangulation for apoint 272 that resides in corresponding triangles 270 and 274 is shownin FIG. 17(b).

Once the triangulation is complete, a determination is made of whetherany of the matching points involved in either of the new triangulationsor the replaced bad point have the undesirable potential for causingtopology deviations (step 264). In one embodiment, this determination ismade as described above with respect to step 206 in FIG. 13. If suchundesirable points are identified in step 264, then the selected badpoint is not replaced, and the triangulations performed with theselected bad point are undone in step 266. Otherwise, the replacementprocess for the selected point is complete, with the selected pointregaining its status as a matching point and being removed from the badpoint queue, and the new triangulations being left in place.

If it was determined in step 254 (FIG. 16) that the selected pointresides in adjoining triangles in the selected regions (as shown in FIG.18(a)), then the corresponding adjoining triangles in each region aremade into a single polygon in step 256. This is achieved by removing thecommon edge that the adjoining triangles share in each of the selectedregions.

Next, a triangulation is performed in step 260 as described above, usingthe vertices of the adjoining triangles in each selected region and theselected bad point. Such a triangulation is illustrated in FIG. 18(b),for the triangles and selected point 294 shown in FIG. 18(a). As shownin FIG. 18(b), the corresponding edges between points P₂ and P₃ andpoints P₂′ and P₃′ were removed before triangulating.

After the triangulation process (step 260), a determination is made instep 264, as described above, to detect the existence of points with thepotential for causing topology deviations. If such points are detectedin step 264, then both triangulations are undone; the edges removed tofacilitate the triangulation are replaced, and the selected bad pointmaintains its bad point status. Otherwise, the bad point regains itsmatching point status and is removed from the bad point queue; the newtriangulations remain in place, and the replacement process (step 214,FIG. 13) is done.

As shown in FIG. 13, once the bad point queue is determined to be empty(step 212), such that it contains no bad points or only bad points thatcannot be triangulated without causing potential for topologicaldeviations, the identification of matching points with potential forcausing topology deviations (step 201, FIG. 12) is done. At this point,the transformation process (step 216, FIG. 12) is initiated.

FIG. 19 illustrates a sequence of operations that are performed in oneembodiment of the present invention to execute the transformation.First, a set of transformation equations are generated in step 360. Aset of transformation equations are generated for each pair ofcorresponding triangles in the selected region of the first map and theselected region of the second map. Each point within a triangle in theselected region in the first map is referred to by a pair of coordinates(x, y), and each point within a triangle in the selected region in thesecond map is identified by a set of coordinates (x′, y′). Each x′coordinate can be expressed by the following equation:

x′=ax+by+c  Equation 12

wherein:

a, b, and c are constants for a pair of corresponding triangles.

Each y′ coordinate can be identified according to the followingequation:

y′=dx+ey+f  Equation 13

wherein:

d, e, and f are constants for a pair of corresponding triangles.

Equations 12 and 13 are known as affine equations, which can be used totransform points within one geometric object and to points in anothergeometric object. In accordance with the present invention, each pair ofcorresponding triangles is employed to solve for a set of the a, b, c,d, e, and f constants, so that a set of affine equations is defined foreach pair of corresponding triangles. Each set of affine equations for atriangle in the selected region of the first map is then used, asdescribed below, for transforming points in the triangle into points ina new map.

The constants in the affine equations for each pair of correspondingtriangles are determined by applying well known algebra techniques. Eachvertex in a triangle in the selected region of the first map and acorresponding triangle vertex in the second map are known. Thecoordinates for each vertex in a pair of corresponding triangles areplugged into the affine equations to obtain 6 affine equations for thepair of corresponding triangles These 6 equations are then employed tosolve for the six a, b, c, d, e, and f constants using standard wellknown algebra techniques.

For example, if a triangle in the selected region of the first map hasvertices {(x₁, y₁), (x₂, y₂), (x₃, y₃)} and a corresponding triangle inthe second selected region of the map has corresponding matching pointvertices {(x₁′, y₁′), (x₂′, y₂′), (x₃′, y₃′)}, then the following affineequations can be set forth for determining the a, b, c, d, e, and fconstants:

x ₁ ′=ax ₁ +by ₁ +c  Equation 14

x ₂ ′=ax ₂ +by ₂ +c  Equation 15

x ₃ ′=ax ₃ +by ₃ +c  Equation 16

y ₁ ′=dx ₁ +ey ₁ +f  Equation 17

y ₂ ′=dx ₂ +ey ₂ +f  Equation 18

y ₃ ′=dx ₃ +ey ₃ +f  Equation 19

These equations can be solved to find the a, b, c, d, e, and f constantsfor the pair of corresponding triangles, thereby making the affineequations for the pair of corresponding triangles known. This process isrepeated, so that affine equations are determined for each pair oftriangles in the selected regions of the first and second maps.

Once the set of affine equations are determined for each pair ofcorresponding triangles in the selected regions of the first and secondmaps (step 360, FIG. 19), a set of arc intersection points in theselected region of the first map are identified in step 362. Each arc inthe selected region in the first map is evaluated to determine whetheris contains a location that is to be identified as an arc intersectionpoint.

An arc intersection point is a location on an arc in the selected regionof the first map at which the arc crosses an edge belonging to twoadjoining triangles. Such a location 370 is illustrated in FIG. 20(a).Arc 376 extends between point 380 in triangle 372 and point 381 intriangle 374. Point 370 is a location on arc 376 that intersects edge382, which is shared by triangles 372 and 374. Prior to transformation,arc 378 does not intersect arc 376. In order to preserve topology whenperforming a transformation to obtain a new map, transformations of arc376 and arc 378 must also not intersect in the new map.

FIG. 20(b) shows the desirable result after transformation, with arc 376being re-shaped into two arc portions 376 a and 376 b that do notintersect arc 378′. However, if no special measures are taken, it ispossible that transformation will result in the situation shown in FIG.20(c). In FIG. 20(c), the end points of transformed arc 376′ are theonly points on arc 376 that were transformed. As a result, thetransformation causes transformed arc 376′ to be falsely intersectedwith transformed arc 378′.

In order to avoid the topology deviation shown in FIG. 20(c), thelocation where arc 376 intersects edge 382 (FIG. 20(a)) is identified asan arc intersection point. As a result, an arc intersection point iscreated at location 370. This shape point will be transformed in step364 (FIG. 19), as described below, to generate point 370′, which will beconnected to arc portions 376 a and 376 b, as shown in FIG. 20(b).

Once all of the arc intersection points in the selected region of thefirst map have been identified, the point transformation in step 364(FIG. 19) is performed. In the point transformation step 364, each point(x, y) in the selected region of the first map is transformed to a point(x′, y′) in a new map, using the affine equations for the triangle inwhich the first map point (x, y) resides. The point transformation stepis also performed on all arc intersection points identified in step 362.As a result, all points of a new map of the selected region aregenerated.

Once the points in the selected region of the first map are alltransformed, the arc connections between the points in the selectedregion of the first map are replicated between the corresponding newlygenerated points in step 366. For example, an arc connected to point (x,y) in the first map is connected to point (x′, y′) in the new map,wherein x′ and y′ are derived from x and y using the affine equations.

A system for performing repositioning in accordance with the presentinvention, as described above, may be implemented in hardware and/orsoftware. In one implementation, the system for repositioning comprisesa dedicated processor including processor instructions for performingthe functions described herein. Circuits may also be developed toperform the functions described herein. In one embodiment, the systemfor performing repositioning is part of a repositioning system. Therepositioning system can be a general purpose computer withrepositioning software. In another implementation, the system forrepositioning includes a plurality of computer executable instructionsfor implementation on a general purpose computer system. Prior toloading into a general purpose computer system, the software may resideas encoded information on a computer readable medium, such as a magneticfloppy disk, magnetic tape, and compact disc read only memory (CD-ROM).

FIG. 21 illustrates a high level block diagram of a general purposecomputer system 410 in which the present inventions system forperforming repositioning is implemented in one embodiment. Inparticular, system 410 can be employed to perform the process stepsdescribed above with reference to FIGS. 5-20.

Computer system 410 contains a processor unit 412 and main memory 414.Processor unit 412 is a single microprocessor in one embodiment, andcontains a plurality of microprocessors for configuring the computersystem 410 as a multi-processor system in another embodiment. Mainmemory 414 stores, in part, instructions and data for execution byprocessor unit 412. When the process steps of the present invention forperforming repositioning are wholly or partially implemented insoftware, main memory 414 stores the executable code during the system'soperation. Main memory 414 can include banks of dynamic random accessmemory (DRAM) as well as high speed cache memory.

Computer system 410 further includes a mass storage device 416,peripheral device(s) 418, input device(s) 420, portable storage mediumdrive(s) 422, a graphics subsystem 424 and an output display 426. Forpurposes of simplicity, the components in computer system 410 are shownin FIG. 21 as being connected via a single bus 428. However, inalternate embodiments, computer system 410 is connected through one ormore data transport means. For example, processor unit 412 and mainmemory 414 may be connected via a local microprocessor bus, and the massstorage device 416, peripheral device(s) 418, portable storage mediumdrive(s) 422, and graphics subsystem 424 may be connected via one ormore input/output (I/O) buses. Mass storage device 416, which may beimplemented with a magnetic disk drive or an optical disk drive, is anon-volatile storage device for storing data and instructions for use byprocessor unit 412. In one embodiment, mass storage device 416 storesthe system software for performing repositioning for purposes of loadingto main memory 14.

Portable storage medium drive 422 operates in conjunction with aportable non-volatile storage medium, such as a floppy disk, to inputand output data and code to and from computer system 410. In oneembodiment, the system software for determining a path is stored on sucha portable medium, and is input to the computer system 410 via theportable storage medium drive 422. Peripheral device(s) 418 may includeany type of computer support device, such as an input/output (I/O)interface, to add additional functionality to the computer system 410.For example, peripheral device(s) 418 may include a network interfacecard for interfacing computer system 410 to a network, a modem, etc.

Input device(s) 420 provide a portion of the user interface for a userof computer system 410. Input device(s) 420 may include an alpha-numerickeypad for inputting alpha-numeric and other key information, or acursor control device, such as a mouse, a trackball, stylus, or cursordirection keys. In order to display textual and graphical information,computer system 410 contains graphics subsystem 424 and the outputdisplay 426. Output display 426 may include a cathode ray tube (CRT)display, liquid crystal display (LCD) or other suitable display device.Graphics subsystem 424 receives textual and graphical information, andprocesses the information for output to output display 426. Outputdisplay 426 can be used to report the results of a repositioning. Thecomponents contained in computer system 410 are those typically found ingeneral purpose computer systems, and are intended to represent a broadcategory of such computer components that are well known in the art. Thesystem of FIG. 21 illustrates one platform which can be used for thepresent invention. Numerous other platforms can also suffice, such asMacintosh-based platforms available from Apple Computer, Inc., platformswith different bus configurations, networked platforms, multi-processorplatforms, other personal computers, workstations, mainframes,navigation systems, and so on.

The foregoing detailed description of the invention has been presentedfor purposes of illustration and description. It is not intended to beexhaustive or to limit the invention to the precise form disclosed, andobviously many modifications and variations are possible in light of theabove teaching. The described embodiments were chosen in order to bestexplain the principles of the invention and its practical application tothereby enable others skilled in the art to best utilize the inventionin various embodiments and with various modifications as are suited tothe particular use contemplated. It is intended that the scope of theinvention be defined by the claims appended hereto.

What is claimed is:
 1. A computer implemented method for forming a firstset of geometric objects, based on a second set of geometric objects anda third set of geometric objects, wherein said second set of geometricobjects contains a second set of matching points and said third set ofgeometric objects contains a third set of matching points, wherein eachmatching point in said second set of matching points corresponds to amatching point in said third set of matching points, said methodcomprising the steps of: (a) identifying matching points in said secondset of matching points and corresponding matching points in said thirdset of matching points that have potential for causing topologydeviations in the first set of geometric objects; and (b) transformingpoints in said second set of geometric objects to obtain points in saidfirst set of geometric objects, based on relationships betweencorresponding matching points in said second set of matching points andsaid third set of matching points other than said matching pointsidentified in said step (a).
 2. The method of claim 1, wherein said step(a) includes the steps of: triangulating said second set of matchingpoints to obtain a second set of triangles; triangulating said third setof matching points to obtain a third set of triangles; and making aninitial identification of matching points in said second set of matchingpoints and said third set of matching points that have potential forcausing topology deviations in said first set of geometric objects. 3.The method of claim 2, wherein said step of making an initialidentification includes the steps of: selecting a pair of triangles,wherein said pair includes a second triangle in said second set oftriangles and a third triangle that is in said third set of trianglesand corresponds to said second triangle; and analyzing said pair oftriangles to determine whether vertices of triangles in said pair arematching points that have potential for causing topology deviations insaid first set of geometric objects.
 4. The method of claim 3, whereinsaid step of analyzing includes the step of: determining whether saidsecond triangle and said third triangle are inverted.
 5. The method ofclaim 3, wherein said step of analyzing includes the step of:determining whether either of said second triangle and said thirdtriangle have acceptable dimensions.
 6. The method of claim 5, whereinsaid step of determining whether either of said second triangle and saidthird triangle have acceptable dimensions, includes the step of:determining whether either said second triangle or said third trianglehas an area less than a minimum acceptable area.
 7. The method of claim5, wherein said step of determining whether either of said secondtriangle and said third triangle have acceptable dimensions, includesthe step of: determining whether either said second triangle or saidthird triangle has a height less than a minimum acceptable height. 8.The method of claim 2, wherein said step (a) further includes the stepof: removing matching status of a set of matching points identified insaid step of making an initial identification.
 9. The method of claim 8,wherein said step (a) further includes the step of: replacing a pointthat had matching status removed in said step of removing matchingstatus.
 10. The method of claim 8, wherein said step (a) furtherincludes the steps of: triangulating said second set of matching pointsto obtain a new second set of triangles, after performing said step ofremoving matching status; triangulating said third set of matchingpoints to obtain a new third set of triangles, after performing saidstep of removing matching status; and making an initial identificationof matching points in said second set of matching points and said thirdset of matching points that have potential for causing topologydeviations in said first set of geometric objects, after performing saidstep of removing matching status.
 11. The method of claim 10, whereinsaid step (b) includes the steps of: generating a set of transformationequations; identifying a set of arc intersection points in said newsecond set of geometric objects; and performing a point transformationon points and arc intersection points in said second set of geometricobjects using said set of transformation equations to obtain points forsaid first set of geometric objects.
 12. The method of claim 11, whereinsaid set of transformation equations includes transformation equationsfor each corresponding pair of triangles in said new second set oftriangles and said new third set of triangles.
 13. The method of claim11, wherein said step (b) further includes the step of: making arcconnections between said points for said first set of geometric objects.14. The method of claim 1, wherein said second set of geometric objectsrepresents a physical area and said third set of geometric objectsrepresents said physical area.
 15. A processor readable storage mediumhaving processor readable code embodied on said processor readablestorage medium, said processor readable code for programming a processorto perform a method for forming a first set of geometric objects, basedon a second set of geometric objects and a third set of geometricobjects, wherein said second set of geometric objects contains a secondset of matching points and said third set of geometric objects containsa third set of matching points, wherein each matching point in saidsecond set of matching points corresponds to a matching point in saidthird set of matching points, said method comprising the steps of: (a)identifying matching points in said second set of matching points andcorresponding matching points in said third set of matching points thathave potential for causing topology deviations in the first set ofgeometric objects; and (b) transforming points in said second set ofgeometric objects to obtain points in said first set of geometricobjects, based on relationships between corresponding matching points insaid second set of matching points and said third set of matching pointsother than said matching points identified in said step (a).
 16. Theprocessor readable storage medium of claim 15, wherein said step (a)includes the steps of: triangulating said second set of matching pointsto obtain a second set of triangles; triangulating said third set ofmatching points to obtain a third set of triangles; and making aninitial identification of matching points in said second set of matchingpoints and said third set of matching points that have potential forcausing topology deviations in the first set of geometric objects. 17.The processor readable storage medium of claim 16, wherein said step ofmaking an initial identification includes the steps of: selecting a pairof triangles, wherein said pair includes a second triangle in saidsecond set of triangles and a third triangle that is in said third setof triangles and corresponds to said second triangle; and analyzing saidpair of triangles to determine whether vertices of triangles in saidpair are matching points that have potential for causing topologydeviations in said first set of geometric objects.
 18. The processorreadable storage medium of claim 17, wherein said step of analyzingincludes the step of: determining whether said second triangle and saidthird triangle are inverted.
 19. The processor readable storage mediumof claim 17, wherein said step of analyzing includes the step of:determining whether either of said second triangle and said thirdtriangle have acceptable dimensions.
 20. The processor readable storagemedium of claim 19, wherein said step of determining whether either ofsaid second triangle and said third triangle have acceptable dimensions,includes the step of: determining whether either said second triangle orsaid third triangle has an area less than a minimum acceptable area. 21.The processor readable storage medium of claim 19, wherein said step ofdetermining whether either of said second triangle and said thirdtriangle have acceptable dimensions, includes the step of: determiningwhether either said second triangle or said third triangle has a heightless than a minimum acceptable height.
 22. The processor readablestorage medium of claim 16, wherein said step (a) further includes thestep of: removing matching status of a set of matching points identifiedin said step of making an initial identification.
 23. The processorreadable storage medium of claim 22, wherein said step (a) furtherincludes the step of: replacing a point that had matching status removedin said step of removing matching status.
 24. The processor readablestorage medium of claim 22, wherein said step (a) further includes thesteps of: triangulating said second set of matching points to obtain anew second set of triangles, after performing said step of removingmatching status; triangulating said third set of matching points toobtain a new third set of triangles, after performing said step ofremoving matching status; and making an initial identification ofmatching points in said second set of matching points and said third setof matching points that have potential for causing topology deviationsin said first set of geometric objects, after performing said step ofremoving matching status.
 25. The processor readable storage medium ofclaim 24, wherein said step (b) includes the steps of: generating a setof transformation equations; identifying a set of arc intersectionpoints in said new second set of geometric objects; and performing apoint transformation on points and arc intersection points in saidsecond set of geometric objects using said set of transformationequations to obtain points for said first set of geometric objects. 26.The processor readable storage medium of claim 25, wherein said set oftransformation equations includes transformation equations for eachcorresponding pair of triangles in said new second set of triangles andsaid new third set of triangles.
 27. The processor readable storagemedium of claim 25, wherein said step (b) further includes the step of:making arc connections between said points for said first set ofgeometric objects.
 28. The processor readable storage medium of claim15, wherein said second set of geometric objects represents a physicalarea and said third set of geometric objects represents said physicalarea.
 29. An apparatus for forming a first set of geometric objects,based on a second set of geometric objects and a third set of geometricobjects, wherein said second set of geometric objects contains a secondset of matching points and said third set of geometric objects containsa third set of matching points, wherein each matching point in saidsecond set of matching points corresponds to a matching point in saidthird set of matching points, said apparatus comprising: a processor;and a processor readable storage medium, in communication with saidprocessor, said processor readable storage medium storing code forprogramming said processor to perform the steps of: (a) identifyingmatching points in said second set of matching points and correspondingmatching points in said third set of matching points that have potentialfor causing topology deviations in the first set of geometric objects;and (b) transforming points in said second set of geometric objects toobtain points in said first set of geometric objects, based onrelationships between corresponding matching points in said second setof matching points and said third set of matching points other than saidmatching points identified in said step (a).
 30. The apparatus of claim29, wherein said step (a) includes the steps of: triangulating saidsecond set of matching points to obtain a second set of triangles;triangulating said third set of matching points to obtain a third set oftriangles; and making an initial identification of matching points insaid second set of matching points and said third set of matching pointsthat have potential for causing topology deviations in said first set ofgeometric objects.
 31. The apparatus of claim 30, wherein said step ofmaking an initial identification includes the steps of: selecting a pairof triangles, wherein said pair includes a second triangle in saidsecond set of triangles and a third triangle that is in said third setof triangles and corresponds to said second triangle; and analyzing saidpair of triangles to determine whether vertices of triangles in saidpair are matching points that have potential for causing topologydeviations in the first set of geometric objects.
 32. The apparatus ofclaim 31, wherein said step of analyzing includes the step of:determining whether said second triangle and said third triangle areinverted.
 33. The apparatus of claim 31, wherein said step of analyzingincludes the step of: determining whether either of said second triangleand said third triangle have acceptable dimensions.
 34. The apparatus ofclaim 33, wherein said step of determining whether either of said secondtriangle and said third triangle have acceptable dimensions, includesthe step of: determining whether either said second triangle or saidthird triangle has an area less than a minimum acceptable area.
 35. Theapparatus of claim 33, wherein said step of determining whether eitherof said second triangle and said third triangle have acceptabledimensions, includes the step of: determining whether either said secondtriangle or said third triangle has a height less than a minimumacceptable height.
 36. The apparatus of claim 30, wherein said step (a)further includes the step of: removing matching status of a set ofmatching points identified in said step of making an initialidentification.
 37. The apparatus of claim 36, wherein said step (a)further includes the step of: replacing a point that had matching statusremoved in said step of removing matching status.
 38. The apparatus ofclaim 36, wherein said step (a) further includes the steps of:triangulating said second set of matching points to obtain a new secondset of triangles, after performing said step of removing matchingstatus; triangulating said third set of matching points to obtain a newthird set of triangles, after performing said step of removing matchingstatus; and making an initial identification of matching points in saidsecond set of matching points and said third set of matching points thathave potential for causing topology deviations in said first set ofgeometric objects, after performing said step of removing matchingstatus.
 39. The apparatus of claim 38, wherein said step (b) includesthe steps of: generating a set of transformation equations; identifyinga set of arc intersection points in said new second set of geometricobjects; and performing a point transformation on points and arcintersection points in said second set of geometric objects using saidset of transformation equations to obtain points for said first set ofgeometric objects.
 40. The apparatus of claim 39, wherein said set oftransformation equations includes transformation equations for eachcorresponding pair of triangles in said new second set of triangles andsaid new third set of triangles.
 41. The apparatus of claim 39, whereinsaid step (b) further includes the step of: making arc connectionsbetween said points for said first set of geometric objects.
 42. Theapparatus of claim 29, wherein said second set of geometric objectsrepresents a physical area and said third set of geometric objectsrepresents said physical area.